Consider the geometry of solar radiation spreading throughout the atmosphere
for concretisation Eq. 1.18 as Fig. 1.7 illustrates. We are presenting the atmosphere
as a model of the plane-parallel and horizontally homogeneous layer. The direction
of the radiation spreading is characterized with the zenith angle ϑ and with the
azimuth ’ counted off an arbitrary direction at a horizontal plane. Set all
coefficients in Eq. 1.18 depending on the altitude (it is completely corresponded
to reality).
Length element dl in the plane-parallel atmosphere is dl ¼ Àdz/cosϑ. The
ground surface at the bottom of the atmosphere is neglected for the present (i.e. it
is accounted that the radiation incoming to the bottom of the atmosphere is not
reflected back to the atmosphere and it is equivalent to the almost absorbing
surface). Within this horizontally homogeneous medium, the radiation field is
also the horizontally homogeneous owing to the shift symmetry (the invariance of
all conditions of the problem relatively to any horizontal displacement). Thus, the
radiance is a function of only three coordinates: altitude z and two angles, defining
direction (ϑ, ’). Hence, Eq. 1.18 could be written as:
dIðz; #; ’Þ
dz
cos # ¼ aðzÞIðz; #; ’Þ À
sðzÞ
4p
ð
2p
0
d’
0
ð p
0
xðz; gÞIðz; #
0
; ’
0
Þ sin #
0 d#
0
(1.19)
where scattering angle g is an angle between directions (ϑ, ’) and (ϑ
0
’
0 ). It is easy
to express the scattering angle through ϑ, ’: to consider the scalar product of the
orts in Cartesian coordinate system and then pass to the spherical coordinates. This
procedure yields the following relation known as Cosine law for the spheroid
triangles cosg ¼ cosϑ cosϑ
0 + sinϑ sinϑ
0 cos(’ – ’
0 ).
To begin with, consider the simplest particular case of transfer Eq. 1.18. Let us
neglect the radiation scattering i.e. the term with the integral. For atmospheric
z
(J 0 ,j 0 = 0) m 0 = arccosJ 0
F 0 = pS
z = z ¥ t = 0
z t
m = arccosJ
j
dl
dz
z = 0 t = t 0
t
®
®
Fig. 1.7 Geometry of
propagation of solar radiation
in the plane parallel
atmosphere
14
1 Radiation in the Earth Atmosphere
for concretisation Eq. 1.18 as Fig. 1.7 illustrates. We are presenting the atmosphere
as a model of the plane-parallel and horizontally homogeneous layer. The direction
of the radiation spreading is characterized with the zenith angle ϑ and with the
azimuth ’ counted off an arbitrary direction at a horizontal plane. Set all
coefficients in Eq. 1.18 depending on the altitude (it is completely corresponded
to reality).
Length element dl in the plane-parallel atmosphere is dl ¼ Àdz/cosϑ. The
ground surface at the bottom of the atmosphere is neglected for the present (i.e. it
is accounted that the radiation incoming to the bottom of the atmosphere is not
reflected back to the atmosphere and it is equivalent to the almost absorbing
surface). Within this horizontally homogeneous medium, the radiation field is
also the horizontally homogeneous owing to the shift symmetry (the invariance of
all conditions of the problem relatively to any horizontal displacement). Thus, the
radiance is a function of only three coordinates: altitude z and two angles, defining
direction (ϑ, ’). Hence, Eq. 1.18 could be written as:
dIðz; #; ’Þ
dz
cos # ¼ aðzÞIðz; #; ’Þ À
sðzÞ
4p
ð
2p
0
d’
0
ð p
0
xðz; gÞIðz; #
0
; ’
0
Þ sin #
0 d#
0
(1.19)
where scattering angle g is an angle between directions (ϑ, ’) and (ϑ
0
’
0 ). It is easy
to express the scattering angle through ϑ, ’: to consider the scalar product of the
orts in Cartesian coordinate system and then pass to the spherical coordinates. This
procedure yields the following relation known as Cosine law for the spheroid
triangles cosg ¼ cosϑ cosϑ
0 + sinϑ sinϑ
0 cos(’ – ’
0 ).
To begin with, consider the simplest particular case of transfer Eq. 1.18. Let us
neglect the radiation scattering i.e. the term with the integral. For atmospheric
z
(J 0 ,j 0 = 0) m 0 = arccosJ 0
F 0 = pS
z = z ¥ t = 0
z t
m = arccosJ
j
dl
dz
z = 0 t = t 0
t
®
®
Fig. 1.7 Geometry of
propagation of solar radiation
in the plane parallel
atmosphere
14
1 Radiation in the Earth Atmosphere